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The E I Abdus Salam International 0) Centre for Theoretical Physics the E I abdus salam .nited navon edumdonal Sent XA0404084 a n'd'cult-l international ."Pn-jon 0) centre international ato.ic energy agency for theoretical physics SCALAR POTENTIAL AND DYONIC STRINGS IN 6D GAUGED SUPERGRAVITI r S. Randjbar-Daemi and E. Sezgin Available at: http: //www. ictp. trieste. it/-pub- of f IC/2004/6 MIFP-03-25 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS SCALAR POTENTIAL AND DI.rONIC STRINGS IN 6D GAUGED SUPERI-TRAVITY S. Randjbar-Daemi The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy and E. Sezgin George P. and Cynthia W. Mitchell Institute for Fundamental Physics, Texas A&M University, College Station, TX 77843-4242, USA. Abstract .In this paper we first give a simple parametrization of the scalar coset manifold of the only known anomaly free chiraJ gauged supergravity in six dimensions in the absence of linear multiplets, namely gauged minimal supergravity coupled to a tensor multiplet, E6 x E7 X U(1)R Yang-Mills multiplets and suitable number of hypermultiplets. We then construct the potential for the scalars and show that it has a unique minimum at the origin. We also construct a new BPS dyonic string solution in which U(1)R X U(1) gauge fields, in addition to the metric, dilaton and the 2-form potential, assume nontrivial configurations in any U(1)R gauged 6D minimaJ supergravity coupled to a tensor multiplet with gauge symmetry G Q U(1). The solution Preserves 14 of the 6D supersymmetries and can be tivially embedded in the anomaly free model, in which case the U(1) activated in our solution resides in E7. MIRAMARE - TRIESTE February 2004 1 Introduction The most symmetric ground state solutions in all of the higher dimensional supergravity theories, as the low energy limit of the superstring theories or the M theory, are the flat 10 dimensional manifolds Rd x f lat torus) and the pp waves. Such backgrounds have 32 real supersymmetries and are not a suitable starting point for building realistic phenomenology. The Calabi Yau compactifications have less supersymmetries but also a lesser degree of uniqueness. There exist many of them with a lot of moduli with unknown potentials. In the gauged minimal supergravity theories in D = 6 this is not the case. In fact in these theories the 6-dimensional flat spaces do not solve the supergravity equations. The most symmetric solution is R4 X S2 [1] which has been shown recently to be the unique maximally symmetric solution of such models 2]. This result has been obtained essentially in the minimal version of such models for which the gauge group is simply U(1), i.e. the D = 6 supersymmetric Einstein Maxwell theory, known as the Salam-Sezgin model in the literature [1]. The model by itself is anomalous but it can be embedded into an anomaly-free model 3] with suitable Yang-Mills and hypermatter sector couplings 4, 5]. The uniqueness of the supersymmetric R4 X S2 solution should be contrasted with the plethora of solutions of Calabi Yau type or the ones with exceptional holonomy groups such as G in higher dimensional low energy string theories. For this reason we consider this property as very interesting and believe that such models deserve further study. This is the aim of the present note. This paper contains two results. First, after giving the multiplet structure of general gauged N = 1 supergravity models in D = 6 we shall concentrate on the hypermatter sector. This sector contains the scalars and fermions which can be in anomaly free representation of the Yang-Mills gauge groups and therefore axe of primary importance in any phenomenological application of such models. So fax we know of only one anomaly free gauged minimal supergravity in D = 6 3] in the absence of linear multiplets 1. For this reason we shall construct in detail the scalar manifold and the potential for the scalars in this particular model. However, our simple parametrization of the quaternionic scalar manifold and the construction of the scalar potential should be applicable in other cases too. The first result of this construction is the observation that the scalar potential admits a unique minimum at the origin. There are no moduli. As we shall see this trivially implies the uniqueness of the R 4 X S2 solution in the anomaly free full fledged model. The second main point of the paper is the construction of new dyonic string solutions. These 'The linear multiplet is a hypermultipIet in which one of the four scalars is dualized to a 4-form potential. In this case, an additional Green-Schwarz counterterm is possible for anomaly cancellation 6], and this may lead to new anomaly free models. 2 :solutions have a rather nontrivial structure and leave 14 of the D = 6 supersymmetries unbro- ken. The search for such solutions is motivated by the general philosophy that they can help us -to study string and field theories from a non-perturbative, semiclassical point of view. 'The choice of the model to be considered here is dictated by anomaly cancellation. It belongs to -a general class of (1,O) supergravity models constructed some time ago 4, 5]. It is based on a six- ,dimensional (1, 0) supergravity theory coupled to a tensor, Yang-Mills and hyper multiplets 3]. At present this is the only known gauged (1, 0) anomaly free supergravity in D = 6 in the -absence of linear multiplets. Its string or M-theory origin is still not quite well understood, although some progress has been made in connecting a subset of the model to M-theory in 11- -dimensions 7], and Horava-Witten type construction in 7-dimensions [8]. Note, however, that in the latter case the R-symmetry group is not gauged i the resulting D = 6 supergravity. Recently, our model has attracted interest in connection with a possibility of obtaining a small cosmological constant in D = 4 9]. These ideas make use of an extension of the old magnetic monopole solution [10] to a situation in which 3-branes are distributed over a transverse S2 [11, 12]. In order for the mechanism to work it is required that the supersymmetry breaking on the brane does not propagate to the bulk. This is the weak point of the scheme for a finite volume 2-manifold, as has been argued in a very general context in 13]. We hope that some other variations of the idea will make a dent on this very important unsolved problem. All the supersymmetric solutions of our model known so far, whether compactifying solutions with a direct product geometry or more involved stringy solutions, use only the UMR gauge fields as part of their ansatz. It seems quite difficult to excite the gauge fields in the non-Abelian E6 x E7 component of the gauge group in search of a supersynnnetric solution. In this paper we report on one such solution in which the gauge field in a U(1) subgroup of E7, in addition to that of the U(1)R factor in the gauge group, is also nonzero. The configuration is quite involved and does not have a 4-dimensional Poincar6 invariance. It has a natural dyonic stringy interpretation similar to the one in [14]. In fact it reduees to the solution in [141 upon setting the E7 gauge field to zero. 'Thus, in our dyonic string solution, in addition to the teELsor fields and the gauge field of U(1)R, a U(1) gauge field embedded in E7 as well as the dilaton field are also active. The only fields which do not participate in the solution are the E6 gauge fields and the hyperscalaxs. The composition of the paper is as follows: In section 2 we give a detailed description of the model with explicit form for the potential in the hyperscalars. In this section we also give the field equations and the supersymmetry transformation rules. In section 3 we show that the -absolute minimum of the scalar potential is at 0 = 0. This fact excludes a solution of the form M4 x K2 with a nonzero vev for any of the gauge fields in E6 x E7 and with any unbroken susy. [n section 3 we discuss the ansatz for the dyonic string with a nonzero vev for the gauge fields 3 in U(1)R x E7 as well as the tensor fields. In section 4 we verify that our ansatz satisfies the field equations and leaves 14 of the original supersymmetries, i.e. one complex susy in 1 + 1 dimensions, unbroken. In the limit of a vanishing E7 field it reduces to the solution found earlier in 14] where only U(1)R field was activated. In this section we also show that our solution has a horizon at = while at r = oo it approaches a cone over a squashed S3 X Minkowski2- The dilaton diverges in both limits. Section contains our conclusions. We give the anomaly polynomial in an Appendix correcting the misprints of 3]. 2 The Model 2.1 Field Content and the Scalar Manifold The six-dimensional gauged supergravity model we shall study involves the following N (1, 0) supermultiplets 2 graviton er oA B- M MN tensor(dilaton) W XA BMN hypermatter V)' YangMills Am AA+ where coordinate basis and tangent space indices are denoted by M, N,... and r, s,..., respec- tively. The antisymmetric tensor potentials, B4N1 give rise to selfdual and anti-selfdual 3-form field strengths.
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